Page 288 - Materials Science and Engineering An Introduction

260 • Chapter 8 / Failure

All brittle materials contain a population of small cracks and flaws that have a
variety of sizes, geometries, and orientations. When the magnitude of a tensile stress at
the tip of one of these flaws exceeds the value of this critical stress, a crack forms and
then propagates, which results in fracture. Very small and virtually defect-free metallic
and ceramic whiskers have been grown with fracture strengths that approach their
theoretical values.

EXAMPLE PROBLEM 8.1

Maximum Flaw Length Computation
A relatively large plate of a glass is subjected to a tensile stress of 40 MPa. If the specific surface
energy and modulus of elasticity for this glass are 0.3 J/m and 69 GPa, respectively, determine
2
the maximum length of a surface flaw that is possible without fracture.

Solution
To solve this problem it is necessary to employ Equation 8.3. Rearranging this expression such
2
that a is the dependent variable, and realizing that s 40 MPa, g s 0.3 J/m , and E 69 GPa,
leads to
2Eg s
a =
ps 2
2
9
(2)(69 * 10 N/m )(0.3 N/m)
=
2 2
6
p(40 * 10 N/m )
-6
= 8.2 * 10 m = 0.0082 mm = 8.2 m

Fracture Toughness
Using fracture mechanical principles, an expression has been developed that relates this
Fracture toughness— critical stress for crack propagation (s c ) and crack length (a) as
dependence on
critical stress for (8.4)
crack propagation K c = Ys c 1pa
and crack length
fracture toughness In this expression K c is the fracture toughness, a property that is a measure of a mate-
rial’s resistance to brittle fracture when a crack is present. K c has the unusual units of
MPa1m or psi1in. (alternatively, ksi1in.). Here, Y is a dimensionless parameter or
function that depends on both crack and specimen sizes and geometries as well as on
the manner of load application.
Relative to this Y parameter, for planar specimens containing cracks that are much
shorter than the specimen width, Y has a value of approximately unity. For example,
for a plate of infinite width having a through-thickness crack (Figure 8.9a), Y 1.0,
whereas for a plate of semi-infinite width containing an edge crack of length a (Figure
8.9b), Y 1.1. Mathematical expressions for Y have been determined for a variety of

crack-specimen geometries; these expressions are often relatively complex.
depends on specimen thickness.
For relatively thin specimens, the value of K c
However, when specimen thickness is much greater than the crack dimensions, K c
plane strain becomes independent of thickness; under these conditions a condition of plane strain
exists. By plane strain, we mean that when a load operates on a crack in the manner